Grade 9

The Pythagorean theorem: practice problems

In a right triangle $a^2 + b^2 = c^2$ . Find a right triangle in the figure and many lengths follow.

Basic, Standard, Advanced: Grade 9 · Term 3

BasicStandardAdvancedMix

The theorem and its converse

Pythagorean theorem

a^2 + b^2 = c^2

Conversely, if $a^2 + b^2 = c^2$ the triangle is right-angled. Common triples: $3:4:5$ , $5:12:13$ , $8:15:17$ .

Special right triangles

A right isosceles triangle has sides in the ratio $1 : 1 : \sqrt{2}$ ; a $30^\circ$ – $60^\circ$ – $90^\circ$ triangle has $1 : 2 : \sqrt{3}$ . An equilateral triangle of side $a$ has height $\frac{\sqrt{3}}{2}a$ .

Applications

Distance: $\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$
Chords: the perpendicular from the centre bisects a chord; a tangent is perpendicular to the radius.
Cuboid diagonal: $\sqrt{a^2 + b^2 + c^2}$
Cones and pyramids: use the right triangle formed by the height and a slant edge.

Shortest paths

Unfold the surface into a net; the shortest path is a straight segment.

Q1BasicGr 9 · Term 3The Pythagorean theorem

Which set of side lengths gives a right triangle?

A $3\text{ cm},\ 4\text{ cm},\ 5\text{ cm}$
B $12\text{ cm},\ 16\text{ cm},\ 21\text{ cm}$
C $12\text{ cm},\ 17\text{ cm},\ 9\text{ cm}$
D $12\text{ cm},\ 5\text{ cm},\ 14\text{ cm}$

Answer

\text{A}

Solution

Choose the set where the square of the longest side equals the sum of the squares of the other two.
$3,\ 4,\ 5$

Q2StandardGr 9 · Term 3The Pythagorean theorem

Find the length of the diagonal of a 8 cm by 12 cm rectangle.

Answer

4\sqrt{13}\text{ cm}

Solution

By Pythagoras,
$\sqrt{8^2 + 12^2} = \sqrt{208} = 4\sqrt{13}$

Q3AdvancedGr 9 · Term 3The Pythagorean theorem

In cuboid ABCD-EFGH, $\mathrm{AB} = 1$ cm, $\mathrm{BC} = 15$ cm and $\mathrm{AE} = 12$ cm. A string runs from A through a point P on edge BF to G. Find the shortest possible length of the string.

Answer

20\text{ cm}

Solution

On the net, AG is the hypotenuse of a right triangle with legs
$1 + 15 = 16,\quad 12$
By Pythagoras,
$\sqrt{16^2 + 12^2} = 20$

Q1BasicGr 9 · Term 3The Pythagorean theorem

Find $x$ in the right triangle.

Q2StandardGr 9 · Term 3The Pythagorean theorem

Find the distance between $\mathrm{A}(2,\ -2)$ and $\mathrm{B}(-3,\ 10)$ .

Q3AdvancedGr 9 · Term 3The Pythagorean theorem

Find the length of a space diagonal of a 11 cm × 2 cm × 10 cm cuboid.

Teleport

Main features

On this page

Topics

Display

The Pythagorean theorem: practice problems

The theorem and its converse

Special right triangles

Applications

Hint

Answer

Solution

Hint

Answer

Solution

Hint

Answer

Solution

Hint

Answer

Solution

Hint

Answer

Solution

Hint

Answer

Solution

Math problem generator

The theorem and its converse

Special right triangles

Applications

Worked examples

Hint

Answer

Solution

Hint

Answer

Solution

Hint

Answer

Solution

Practice problems

Hint

Answer

Solution

Hint

Answer

Solution

Hint

Answer

Solution

Related topics

Square roots

Similar figures