Pythagorean Theorem — Fiche d'exercices de Geometry

Grades 8–10 Difficulté moyenne 10 problèmes

The Pythagorean theorem states that in a right triangle with legs $a$ and $b$ and hypotenuse $c$ , $a^2 + b^2 = c^2$ . It applies only to right triangles — triangles containing a $90°$ angle.

The hypotenuse is always the longest side, opposite the right angle. When finding a leg, rearrange to $a^2 = c^2 - b^2$ . Always take the positive square root for side lengths.

The converse helps determine if a triangle is right: if $a^2 + b^2 = c^2$ , the triangle is a right triangle. This is useful in construction and navigation.

Compétences travaillées

Identifying hypotenuse and legs
Finding missing side lengths
Applying the theorem to word problems
Using the converse to test for right triangles

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Fiche d'exercices : Pythagorean Theorem

Consignes : Résous chaque problème avec soin. Montre clairement tout ton travail. Écris ta réponse finale dans l'espace prévu ou sur une feuille séparée comme indiqué.

1.A right triangle has legs of lengths 6 cm and 8 cm. Find the length of the hypotenuse.

Pythagorean Theorem — Fiche d'exercices (suite)

2.A 13-foot ladder is leaning against a wall. The base of the ladder is 5 feet from the wall. How high up the wall does the ladder reach?
3.Find the missing side length of a right triangle with a hypotenuse of 25 inches and one leg of 7 inches.

Pythagorean Theorem — Fiche d'exercices (suite)

4.A rectangular field is 40 meters long and 30 meters wide. What is the distance from one corner to the opposite corner?
5.The legs of a right triangle are 9 cm and 12 cm. What is the perimeter of the triangle?

Pythagorean Theorem — Fiche d'exercices (suite)

6.A television screen measures 32 inches diagonally and 24 inches in height. What is the width of the screen?
7.A right triangle has a hypotenuse of 17 feet and one leg of 15 feet. Find the length of the other leg.

Pythagorean Theorem — Fiche d'exercices (suite)

8.A ramp rises 3 feet over a horizontal distance of 4 feet. What is the length of the ramp?
9.Two friends start at the same point. One walks 5 miles north, the other walks 12 miles east. How far apart are they?

Pythagorean Theorem — Fiche d'exercices (suite)

10.A right triangle has legs of lengths $x$ and $x+1$ , and a hypotenuse of length 5. Find the value of $x$ .

Corrigé

1.

$c = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10$ cm

Réponse finale : 10 cm
2.

$h = \sqrt{13^2 - 5^2} = \sqrt{169 - 25} = \sqrt{144} = 12$ feet

Réponse finale : 12 feet
3.

$b = \sqrt{25^2 - 7^2} = \sqrt{625 - 49} = \sqrt{576} = 24$ inches

Réponse finale : 24 inches
4.

$d = \sqrt{40^2 + 30^2} = \sqrt{1600 + 900} = \sqrt{2500} = 50$ meters

Réponse finale : 50 meters
5.

Hypotenuse $= \sqrt{9^2 + 12^2} = \sqrt{81 + 144} = \sqrt{225} = 15$ cm. Perimeter $= 9 + 12 + 15 = 36$ cm

Réponse finale : 36 cm
6.

$w = \sqrt{32^2 - 24^2} = \sqrt{1024 - 576} = \sqrt{448} \approx 21.2$ inches

Réponse finale : 21.2 inches

Corrigé (suite)

7.

$b = \sqrt{17^2 - 15^2} = \sqrt{289 - 225} = \sqrt{64} = 8$ feet

Réponse finale : 8 feet
8.

$r = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5$ feet

Réponse finale : 5 feet
9.

$d = \sqrt{5^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13$ miles

Réponse finale : 13 miles
10.

$x^2 + (x+1)^2 = 5^2$ ; $x^2 + x^2 + 2x + 1 = 25$ ; $2x^2 + 2x - 24 = 0$ ; $x^2 + x - 12 = 0$ ; $(x+4)(x-3)=0$ ; $x=3$ (positive solution)

Réponse finale : 3

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Erreurs fréquentes à éviter

Adding legs when the hypotenuse is needed (should subtract)
Forgetting the square root at the end
Applying the theorem to non-right triangles
Using the wrong side as the hypotenuse

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