Performing calculations on real numbers, also using a
calculator, applying the laws of mathematical operations when
manipulating algebraic expressions, and using these skills to
solve problems in real and theoretical contexts.
Use and creation of information.
Interpreting
and manipulating information presented in the text, both
mathematical and popular science, as well as in the form of
charts, diagrams, tables.
Using
mathematical language to create mathematical texts, including
description of reasoning and justification of conclusions, as
well as to present data.
Use and interpretation of representation.
Applying and operating mathematical objects,
interpreting mathematical concepts.
Selection and creation of mathematical models for
solving practical and theoretical problems.
Creating auxiliary mathematical objects on the basis of
existing, to conduct arguments or solve a problem.
Indicating
the necessity or the possibility of modifying the mathematical
model in cases requiring special reservations, additional
assumptions, consideration of specific conditions.
Reasoning and argumentation.
Carrying out reasoning, including several stages, providing
arguments justifying the correctness of reasoning,
distinguishing evidence from an example.
Seeing regularities, similarities and analogies, formulating
conclusions based on them and justifying their correctness.
Choosing arguments to justify the correctness of
problem solving, creating a series of arguments that guarantee
correctness of the solution and effectiveness in finding
solutions to the problem.
Applying and creating strategies when solving tasks, also in
unusual situations.
Teaching contents - detailed requirements
I.
Real numbers
Standard level. A pupil:
performs operations (addition, subtraction, multiplication, division,
exponentiation, extracting roots, logarithm) in a set of real numbers;
carries simple proofs about the divisibility of integers and division
remainders, no more difficult than:
(a) proof that the product of four consecutive natural numbers is
divisible by $24$ of ;
(b) proof that if a number when dividing by $5$ gives the remainder of
$3$, then its third power when dividing by $5$ gives the remainder of
$2$;
applies properties of roots of any degree, including odd degree roots
of negative numbers;
applies the relationship between roots and exponents, and laws of
exponents and roots;
applies
monotonicity properties of exponentiation, in particular:
if $x < y$ and
$a>1$, then
$a^x<a^y$ , and if
$x < y$ and
$0<a<1$, then
$a^x>a^y$ ;
uses the concept of real interval, marks intervals on a number line
;
applies geometrical and algebraic interpretation of absolute value,
solves equations and inequalities of the type: $\left|x + 4\right| = 5$, $\left|x -
2\right| < 3$, $\left|x+3\right| \geq 4$;
uses properties of exponent and root properties in practical
situations, including calculating compound interest, investment
returns, and loan costs;
uses the relation between logarithms
and exponents, uses formulas for a logarithm of the product, a
logarithm of the quotient and a logarithm of a power
.
A pupil meets the requirements specified for the standard level, and
also
1R. applies the formula to replace the base of the logarithm.
adds,
subtracts and multiplies one and many variables polynomials;
takes a
common monomial out of an algebraic sum;
factorizes polynomials by grouping
terms, in no more difficult cases than factorizing the polynomial
$W(x)=2x^3-\sqrt{3}x^2+4x-2\sqrt{3}$;
finds
integer rootsof a polynomial with integer coefficients;
dzieli z resztą wielomian jednej zmiennej $W(x)$ przez dwumian postaci
$x-a$;
adds
and subtracts rational expressions, in cases not more difficult than:
$\frac{1}{x+1}-\frac{1}{x}$,
$\frac{1}{x}+\frac{1}{x^2}+\frac{1}{x^3}$,
$\frac{x+1}{x+2}+\frac{x-1}{x+1}$.
A pupil meets the requirements specified for the standard level, and
also
1R.
finds
integer and rational polynomial roots with integer coefficients
2R.
uses
the basic properties of the Pascal triangle and the following
properties of the binomial coefficient: $\binom{n}{0}=1$,
$\binom{n}{1}=n$, $\binom{n}{n-1}=n$, $\binom{n}{k}=\binom{n}{n-k}$,
$\binom{n}{k}+\binom{n}{k+1}=\binom{n+1}{k+1}$;
3R. uses formulas: $a^3+b^3$, $(a+b)^n$ and $(a-b)^n$.
III.
Equations and inequalities.
Standard level. A pupil:
transforms
equations and inequalities in an equivalent manner;
interprets
equations and inequalities that are identities or contradictions;
solves linear inequalities with one unknown;
solves quadratic
equations and inequalities;
solves compound equations that lead to
a quadratic equation, in particular to an equation of the form
$ax^4+bx^2+c=0$;
solves polynomial equations of the
form $W (x) = 0$ for polynomials given in factorized form or
polynomials that can be factorized by grouping terms;
solves rational equations of the form $\frac{V (x)}{W (x)} = 0$, where
polynomials $V(x)$ and $W(x)$ are written in a factorized form.
A pupil meets the requirements specified for the standard level, and
also:
III. 1R.
solves polynomial
inequalities such as: $W(x)> 0$, $W(x)\geq 0$, $W(x)<0$
for polynomials given in a factorized form or polynomials that can
be factorized by taking out a common factor or by grouping terms;
III. 2R.
solves rational equations and inequalities, no more difficult than
$\frac{x+1}{x(x-1)}+\frac{1}{x+1} \geq \frac{2x}{(x-1)(x+1)}$;
III. 3R.
uses Vièta's
formulas for quadratic equations;
III. 4R.
solves equations and inequalities with an absolute value of no more
difficult than: $\left|x + 2\right|+ 3\left|x -1\right| = 13$,
$\left|x + 2\right|+ 3\left|x -1\right| < 11$;
III. 5R.
analyzes linear equations and inequalities with parameters and
quadratic equations and inequalities with parameters, in particular,
determines the number of solutions depending on parameters, gives
the conditions under which solutions have the desired property, and
determines solutions depending on parameters.
IV.
Systems of equations.
Standard level. A pupil:
solves
systems of linear equations with two unknowns, presents geometric
interpretation of conssistent independent, consistent dependent and
inconsistent systems of equations;
uses
equation systems to solve word problems;
uses substitution method to solve systems of
equations, one of which is linear and the other is quadratic:
$\begin{cases} ax + by = e \\ x^2 + y^2 + cx + dy = f \end{cases}$
or $\begin{cases} ax + by = e \\ y = cx^2 +
dx + f \end{cases}$ .
A pupil meets the requirements specified for the standard level, and
also
IV. 1R.
solves systems
of quadratic equations of the form $\begin{cases} x^2 + y^2 + ax +
by = c \\ x^2 + y^2 + cx + dy = f \end{cases}$ .
V.
Functions.
Standard level. A pupil:
defines functions as an unambigues assignment using a verbal
description, table, graph, formula (also with different patterns at
different intervals);
evaluates
output of a function based on given input and algebraic formula of a
function;
reads and interprets
the values of functions defined by tables, charts, formulas etc., also
in case of multiple use of the same source of information or several
sources at the same time;
reads from the graph of functions: domain, range, zeros, monotonicity
intervals, intervals in which the function reaches values larger (or
not smaller) or smaller (or not larger) than a given number, the
largest and smallest values of a function (if any) in a given closed
interval and inputs for which the function has the largest and
smallest values;
interprets
the coefficients in a formula of the linear function;
determines
the formula of a linear function based on information about its graph
or its properties;
sketches a
graph of a quadratic function given by a formula;
interprets
the coefficients appearing in the formula of the quadratic function in
general, canonical and product form (if it exists);
determines
a formula for a quadratic function based on information about the
function or its graph;
determines
the largest and smallest values of the quadratic function in a closed
interval;
uses
the properties of linear and quadratic functions to interpret
geometric, physical, etc. issues, also embedded in a practical
context;
based on the graph of a function $y=f(x)$
sketches graphs of functions $y=f(x-a)$, $y=f(x)+b$, $y=-f(x)$,
$y=f(-x)$;
uses
the function $f(x)=\frac{a}{x}$, including its graph, to describe and
interpret issues related to inversely proportional quantities, also in
practical applications;
uses
exponential and logarithmic functions, including their graphs, to
describe and interpret issues related to practical applications.
A pupil meets the requirements specified for the standard level, and
also
V. 1R.
based
on the graph of the function $y=f(x)$ draws a graph of the function
$y=\left| f(x) \right|$;
V. 2R.
uses composition of
functions;
V. 3R.
proves
the monotonicity of the function given by the formula, as in the
example: prove that the function $f(x)=\frac{x-1}{x+2}$
is monotonic in the interval $(-\infty , -2)$.
VI.
Sequences.
Standard level. A pupil:
evaluates
terms of a sequence defined by a closed formula;
evaluates
a few initial terms of a sequence given by recursion, like in
following examples:
a) $\begin{cases} a_1 = 0,001 \\
a_{n+1}=a_{n}+\frac{1}{2}a_{n}(1-a_{n}) \end{cases}$ ,
b) $\begin{cases} a_1 = 1 \\ a_2 = 1
\\a_{n+2}=a_{n+1}+a_{n} \end{cases}$ .
in
simple cases, examines whether a given sequence is increasing or
decreasing;
checks
if the given sequence is arithmetic or geometric;
uses
a formula for $n$-th term and the sum of $n$ initial terms of the
arithmetic sequence;
uses a formula for $n$-th term and the sum of $n$ initial
terms of the geometric sequence;
uses
properties of sequences, including arithmetic and geometric ones, to
solve tasks, also embedded in a practical context.
A pupil meets the requirements specified for the standard level, and
also
VI. 1R.
;
calculates limits of sequences using limits of the sequences such as
$ \ frac {1} {n} $, $ \ sqrt [n] {a} $ & nbsp; and theorems on the
limits of sum, difference, product and quotient of convergent
sequences, as well as theorems on three sequences;
VI. 2R.
recognizes converging geometrical series and calculates their sum.
VII.
Trigonometry.
Standard level. A pupil:
uses
the definitions of sine, cosine and tangent for angles from $0^\circ $
to $ 180^\circ$, in particular, sets the values of trigonometric
functions for angles $30^\circ$, $45^\circ$, $60^\circ$;
finds
approximate values of trigonometric functions using tables or a
calculator;
finds
the approximate angle size using trigonometric tables or a calculator
if the value of the trigonometric function is given;
uses
theorems of sines and cosines and the formula
$P=\frac{1}{2}absin\gamma$ to calculate the are of a triangle;
calculates
angle sizes of a triangle and lengths of its sides with the
appropriate data (solves triangles).
A pupil meets the requirements specified for the standard level, and
also
VII. 1R.
uses an arc measure, converts an angle measure from degrees to
radianse, and vice versa;
VII. 2R.
uses graphs of trigonometric functions: sine, cosine, tangent;
VII. 3R.
uses periodicity of trigonometric functions;
VII. 4R.
uses
reduction formulas for trigonometric functions
VII. 5R.
uses
formulas for sine, cosine and tangent of sum and difference of
angles, as well as for trigonometric functions of doubled angles;
VII. 6R.
solves trigonometric equations and inequalities
of no more difficulty than in the examples: $4\cos2x\cos5x =
2\cos7x+1$, $2\sin^2x \leq 1$.
VIII.
Planimetry
Standard level. A pupil:
determines
radii and diameters of circles, lengths of chords of circles and
tangent segments, including using Pythagoras' theorem;
recognizes
acute, right and obtuse triangles of given side lengths (using, if
necessary, the inverse theorem to Pythagoras' theorem and cosine
theorem); uses the statement: in the triangle opposite the larger
internal angle lies the longer side;
recognizes
regular polygons and uses their basic properties;
uses
the properties of angles and diagonals in rectangles, parallelograms,
rhombuses and trapeziums;
applies properties of inscribed and central angles;
applies
formulas of the area of the circle sector and the length of the arc of
a circle;
applies
theorems: Thales's theorem, inverse to Thales theorem, theorem about
angle bisector in a triangle and the angle between the tangent and the
chord;
uses the similarity
rules for triangles;
uses
relationships between areas of similar figures;
indicates
special points in a triangle: incenter, circumcenter, orthocenter, centroid, and uses their
properties;
uses
trigonometric functions to determine lengths of segments in a plane
figure and to calculate the area of a figure;
carries
out geometric proofs.
A pupil meets the requirements specified for the standard level, and
also
VIII.1R.
applies
the properties of quadrangles inscribed in a circle and described on
a circle.
IX.
Analytical
geometry on the Cartesian plane.
Standard level. A pupil:
recognizes
the relative position of lines on a plane based on their equations,
including finding a common point of two lines, if one exists;
recognizes
the relative position of lines on a plane based on their equations,
including finding a common point of two lines, if one exists;
calculates
the distance between two points in a coordinate system;
uses the
circle equation $(x-a)^2+(y-b)^2=r^2$;
calculates the distance of
a point from a straight line;
finds
common points of the straight line and the circle as well as the
straight line and the parabola being a graph of the quadratic
function;
determines
the images of circles and polygons in axial symmetries about axes of
the coordinate system, central symmetry (with a center at the origin).
A pupil meets the requirements specified for the standard level, and
also
IX. 1R.
uses the equation
of a circle in general form;
IX. 2R.
finds common points
of two circles;
IX. 3R.
knows the concept of a vector and calculates its coordinates and
length, adds vectors and multiplies the vector by a number, both of
these actions are performed both analytically and geometrically.
X.
Stereometry.
Standard level. A pupil:
recognizes
the relative position of lines in space, in particular perpendicular
lines that do not intersect;
uses
the concept of the angle between the straight line and the plane and
the concept of the dihedral angle between the half planes;
recognizes
angles between segments (e.g. edges, edges and diagonals) and angles
between faces in prisms and pyramids, calculates measures of these
angles;
recognizes
in cylinders and cones the angle between segments and the angle
between segments and planes (e.g. cone opening angle, slant angle),
calculates measures of these angles;
determines
what figure is a given cross-section of a given cuboid;
calculates
the volume and surface area of prisms, pyramids, cylinder, cone and
sphere, also using trigonometry and theorems known;
uses
the relationship between the volumes of similar solids.
A pupil meets the requirements specified for the standard level, and
also
X. 1R.
knows
and applies the theorem about a line perpendicular to the plane and
about three perpendiculars;
X. 2R.
determines
the cross-sections of the cube and normal pyramids and calculates
their fields, also using trigonometry.
XI.
Combinatorics.
Standard level. A pupil:
counts
objects in simple combinatorial situations;
counts
objects using multiplication and addition rules (also together) for
any number of actions in situations not more difficult than:
a)
finding
the number of four-digit odd positive integers such that exactly one
digit 1 and exactly one digit 2 appear in their decimal notation,
b)
finding
the number of positive four-digit even integers such that exactly one
digit 0 and exactly one digit 1 appear in their decimal notation;
A pupil meets the requirements specified for the standard level, and
also
XI. 1R.
calculates the number of possible situations that meet certain
criteria, using the rule of multiplication and addition (also
together) and formulas for the number of: permutations, combinations
and variations, also in cases requiring consideration of a complex
model of counting elements;
XI. 2R.
uses the binomial coefficient and its properties in solving
combinatorial problems.
XII.
Probability and
statistics.
Standard level. A pupil:
calculates the
probability in a classic model;
uses the centile scale;
calculates
the arithmetic and weighted average, finds the median and dominant;
calculates
the standard deviation of a data set (also in the case of
appropriately grouped data), interprets this parameter for empirical
data;
calculates
the expected value, e.g. when determining the amount of winnings in
simple games of chance and lotteries.
A pupil meets the requirements specified for the standard level, and
also
XII. 1R.
calculates
the conditional probability and uses the Bayes formula, practically
applies the total probability theorem;
XII. 2R.
uses the Bernoulli scheme.
XIII.
Optimization and calculus
Standard level. A pupil:
A
pupil solves optimization tasks in situations that can be described by
a quadratic function.
A pupil meets the requirements specified for the standard level, and
also
XIII. 1R.
calculates
function boundaries (including one-sided);
XIII. 2R.
uses
the Darboux property to justify the existence of a function zero and
to find the approximate value of a zero;
XIII. 3R.
applies
the definition of a derivative of function, gives a geometric and
physical interpretation of the derivative;
XIII. 4R.
calculates
the derivative of a power function with a real exponent and
calculates the derivative using theorems on the derivative of sum,
difference, product, quotient and composite function;
XIII. 5R.
uses a
derivative to study the monotonicity of a function;
XIII.6R.
solves
optimization problems using a derivative.
Conditions and manner of
implementation.
Correlation.
Because of the usefulness of mathematics and its applications in school
teaching Physics, computer science, geography and chemistry it is
advised to implement the teaching content specified in sections: I point
9 (logarithms) and, if possible, V point 14, V point 1 (concept of
function) and V point 5 (linear functions) in the first half of the
first year, and the content of teaching specified in sections: V point
11 (quadratic functions) and V point 13 (inverse proportionality) no
later than the end of first grade. The content of teaching specified in
section VI point 2 (calculating the initial words of the recursively
specified strings) can be performed in correlation with the same problem
of the core curriculum in computer science.
Mathematical symbols.
A pupils should use commonly accepted
symbols for numerical sets, in particular: for integers $\mathbb{Z}$,
for rational numbers - $\mathbb{Q}$, for real numbers - $\mathbb{R}$.
The symbol $C$ for the set of integers can lead to confusion and should
be avoided.
Intervals.
The A pupil should use the intervals to describe the set of
solutions of an inequality. It is worth emphasizing that the most
important thing about the answer is its correctness. For example,
resolving the inequality $ x^2-9x+20>0$ can be credited to any of the
following ways:
the
inequality is satisfied by numbers $x$ that are less than 4 or greater
than $5$;
all numbers $x$ less than $4$ and all numbers $x$ greater than $5$
satisfy the inequality;
$x<4$ or $x>5$;
$x\in (-\infty, 4)$ or $x\in (5, \infty)$;
$ x\in (-\infty, 4) \cup (5, \infty)$.
Logarithm applications.
When
teaching logarithms, it is worth highlighting their applications. in
explaining natural phenomena. In nature, processes whose logarithmic
function describes are common. This happens when in a certain period of
time a given quantity always increases (or decreases) with a constant
fold. The following sample problems illustrate the use of logarithms.
Problem 1.
The Richter scale is used to determine the strength of earthquakes.
This force is described by the formula $R=\log \frac{A}{A_0}$, where
$A$ is the quake amplitude expressed in centimeters, $A_0=10^{-4}$ cm
is a constant, called the reference amplitude. On May 5, 2014, a $6.2$
rich magnitude earthquake occurred in Thailand. Calculate the
earthquake amplitude of the land in Thailand.
Problem 2.
The patient took a
dose of $100$ mg of the drug. The mass of this drug remaining in the
body after time $t$ is determined by the relationship $M(t)=a\cdot
b^t$. After five hours, the body removes $30$% of the drug. Calculate
how much medicine will remain in the patient's body after a day.
Vertical form.
When
dealing with square polynomials it should be emphasized vertical form of
a quadratic function and the resulting properties. It should be noted
that the formulas for the roots of quadratic aquation and the
coordinates of the top of the parabola are only conclusions from it. It
is worth emphasizing that many issues associated with the quadratic
function can be solved directly from the the vertex form, without
mechanical application of formulas. In particular, the vertical form
allows you to find the smallest or largest value of a quadratic
function, as well as the axis of symmetry of its plot.
Composite functions and
inverse functions.
The definition of a composite function appears only in the advanced
level, but in the standard level a A pupil is expected to be able to use
data from several sources simultaneously. However, this does not require
any formal introduction of composition or inverse function.
Equivalent transformations .
When
solving equations and inequalities, it should be noted that instead the
method of equivalent transformations, you can use the inference method
(ancient analysis method). After determining the potential set of
solutions, it is checked which of the determined values are the
solutions. In many situations, it is not worth demanding equivalent
transformations when the inference method leads to quick results. In
addition, A pupils should know that the legitimate method of proof is
equivalent transformation of the thesis.
Applications of algebra.
A
prerequisite for successful math teaching process is efficient using
algebraic expressions. Algebraic methods can often be used in geometric
situations and vice versa - geometric illustration allows a better
understanding of algebraic issues.
Sequences
This issue should be discussed so that A pupils realize that there are
others besides arithmetic and geometric sequences. Similarly, it should
be emphasized that apart from non-decreasing, growing, non-growing,
decreasing and constant sequences, there are also ones that are not
monotonic. It is worth noting that some sequences describe the dynamics
of processes occurring in nature or society. For example, given in
section VI point 2 lit. and the string describes the spread of the rumor
($a_n$ indicates how many people have heard of the rumor). A similar
model can be used to describe the spread of the epidemic.
Planimetry
Solving
classic geometric problems is an effective way to shape mathematical
awareness. As a result, A pupils who solve construction problems acquire
skills in solving geometric problems of various types, for example, A
pupils can easily acquire the properties of circles inscribed in a
triangle or quadrangle, if they can construct these figures. Teaching
geometric constructions can be carried out in a classic way, using a
ruler and a compass, or you can use specialized computer programs, such
as GeoGebra.
Stereometry.
Spatial imagination is particularly developed during the implementation
of teaching content from stereometry. Using solid models, as well as the
ability to draw their projections, will greatly facilitate the
determination of different sizes in solids. Cross-section analysis of a
tetrahedron and a cube can be very informative; particularly valuable is
the answer to the question: what a cross-section can be. Experience
teaches that, for example, the question of the existence of a cube
cross-section, which is a trapezium but not isosceles, can cause trouble
for many A pupils.
Binomial expansion.
It is important to emphasize the importance of the binomial coefficient
$\binom{n}{k}$ in combinatorics when teaching the formula for $(a +
b)^n$. It is also worth to write it in the form
$\binom{n}{k}=\frac{n(n-1)\cdot ... \cdot(n-k + 1)}{1 \cdot 2 \cdot ...
\cdot (k-1) \cdot k}$, because in this form its interpretation is more
visible and easier to calculate for small $k$.
Probability.
In the future, A pupils will deal with issues related to randomness that
occur in various areas of life and science, for example, when analyzing
surveys, issues in economics and financial market research or in natural
and social sciences. It is worth mentioning the paradoxes in the theory
of probability, which show typical errors in reasoning and discuss some
of them. It is also worth conducting experiments with A pupils, e.g. an
experiment in which A pupils save a long string of heads and tails
without tossing-up coins, and then save the string of heads and heads
resulting from random coin tosses. Misconceptions about randomness
usually suggest that there should not be long sequence of tails (or
heads), when in reality such long sequence of tails (or heads) occur.
Discussing the basic expected value does not require the introduction of
the concept of random variable. It is advisable to use an intuitive
understanding of the expected value of profit or to determine the number
of objects that meet certain properties. In this way, the A pupil has
the opportunity to see the relationship of probability with everyday
life, also has the chance to shape the ability to avoid risky behaviors,
e.g. in financial decisions
In the advanced level, it is important to make A pupils aware that the
theory of probability is not limited to the classical scheme and the
combinatorics used there. A good illustration are examples of using the
Bernoulli scheme for a large number of attempts.
Proofs.
Idependent carrying out proofs by A pupils develops skills such as
logical thinking, precise expression of thoughts and the ability to
solve complex problems. Command allows you to improve your ability to
choose the right arguments and construct the right reasoning. One of the
methods to develop the skill of proving is to analyze the evidence of
the theorems learned. In this way, you can teach what a properly
conducted piece of evidence should look like. Being able to formulate
correct reasoning and justifications is also important outside of
mathematics. Below is a list of statements whose evidence the A pupil
should know.
Theorems, proofs -
standard level.:
The
existence of infinitely many primes.
Proof of irrationality
of numbers: $\sqrt{2}$ , $\log_2{5}$ itp.
Formulas
for zeros of the quadratic trinomial.
Basic
properties of powers (with integer and rational exponents) and
logarithms.
Theorem about division with the remainder of the polynomial by a
binomial of the form $x-a$ together with recursive formulas for the
quotient and remainder coefficients (Horner's algorithm) - proof can
be carried out in a special case, e.g. for a fourth-degree polynomial.
Closed formulas for $n$-th term and the sum of $n$ initial
terms of the arithmetic and geometric sequence.
Theorem on angles
in a circle:
1)
Central
angle is twice any inscribed angle subtended by the same arc;
2) Two angles inscribed in the same circle are congruent if and only
if they are subtended by the arc of the same length.
Theorem
of segments in a right triangle:
If a segment $CD$ is the height of a right triangle $ABC$ with the
right angle $ACB$, then $\left|AD\right|
\cdot\left|BD\right|=\left|CD\right|^2$, $\left|AC\right|^2=
\left|AB\right|\cdot\left|AD\right|$ oraz $\left|BC\right|^2=
\left|AB\right|\cdot\left|BD\right|$.
Triangle angle bisector theorem:
If
a line $CD$ is the angle bisector of the angle $ACB$ in a triangle
$ABC$ and the point $D$ lies on the side $AB$, then
$\frac{|AD|}{|BD|}=\frac{|AC|}{|BC|}$.
The formula for the area of a triangle
$P=\frac{1}{2}ab \sin \gamma$
.
Sine theorem.
Cosine
theorem and the theorem inverse to Pythagoras's theorem.