21 problems graded by difficulty (★ easy, ★★ medium, ★★★ hard). Try first, without looking at the solutions!
1 Hypotenuse of a 6-8 triangle ★★★
A right triangle has legs 6 cm and 8 cm. Find the length of the hypotenuse.
2 A longer triple ★★★
The legs of a right triangle measure 7 in and 24 in. How long is the hypotenuse?
3 Find a missing leg ★★★
A right triangle has hypotenuse 26 m and one leg 10 m. Find the other leg.
4 True or false? ★★★
Mia says: “A right triangle with legs 4 cm and 5 cm has a hypotenuse of 9 cm.” Is she right? Justify.
5 Three hypotenuses ★★★
Find the hypotenuse of each right triangle given its legs.
- \(a = 8\), \(b = 15\)
- \(a = 9\), \(b = 40\)
- \(a = 12\), \(b = 35\)
6 A hypotenuse that is not a whole number ★★★
The legs of a right triangle are 3 ft and 5 ft. Give the exact hypotenuse, then round it to the nearest hundredth.
7 Right triangle or not? ★★★
Decide whether each triangle is a right triangle.
- Sides 9 in, 12 in, 15 in
- Sides 7 in, 8 in, 10 in
8 The leaning ladder ★★★
A 15 ft ladder leans against a vertical wall. Its foot is 9 ft from the base of the wall. How high up the wall does the ladder reach?
9 A shortcut across a field ★★★
A rectangular practice field measures 24 m by 32 m. Jordan runs along the diagonal instead of along two sides. How many meters does the shortcut save?
10 Height of an isosceles triangle ★★★
An isosceles triangle has two sides of 13 cm and a base of 10 cm. Find its height to the base, then its area.
11 Distance between two points ★★★
Find the distance between \(A(1, 2)\) and \(B(9, 17)\).
12 Distance with negative coordinates ★★★
Find the distance between \(P(-4, 3)\) and \(Q(8, -2)\).
13 A triangle with decimal sides ★★★
A triangle has sides 1.5 cm, 2 cm and 2.5 cm. Is it a right triangle? If so, which side is opposite the right angle?
14 Flight of a drone ★★★
A drone takes off, flies 120 m due east, then 50 m due north. (a) How far is it from its starting point in a straight line? (b) How long is the whole trip if it then flies straight back?
15 Will the rod fit in the box? ★★★
A shipping box is 12 in by 16 in by 21 in. (a) Find the length of its space diagonal. (b) Can a straight rod 28 in long fit inside? What about a 30 in rod?
16 The broken tree ★★★
A 16 ft tree snaps at a point \(x\) feet above the ground. The top bends over and touches the ground 8 ft from the base of the trunk. At what height did the tree break?
17 Area and perimeter from a hypotenuse ★★★
A right triangle has hypotenuse 25 cm and one leg 7 cm. Find the other leg, the perimeter and the area.
18 A right angle in the coordinate plane ★★★
Let \(A(0, 0)\), \(B(4, 3)\) and \(C(-3, 4)\). Compute \(AB^2\), \(AC^2\) and \(BC^2\). Is triangle \(ABC\) a right triangle? Where is the right angle?
19 Diagonal of a square ★★★
The diagonal of a square is 14 cm. Find the exact side length, then its approximate value to the nearest hundredth, and the area of the square.
20 Height of an equilateral triangle ★★★
An equilateral triangle has side 10 cm. Find its height and its area, rounded to the nearest hundredth.
21 Distance in 3D coordinates ★★★
In three-dimensional space, find the distance between \(P(1, 2, 3)\) and \(Q(3, 6, 7)\).
Test yourself: quick challenge for Grade 8
Speed drill for Grade 8: how many in 60 seconds?
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