Practice Leg and altitude theorems
In a right-angled triangle, the leg and altitude theorems relate the legs, the altitude and the segments of the hypotenuse.
They help you find missing lengths.
Exercises with solutions
In a right-angled triangle the altitude h splits the hypotenuse c into two sections p and q. The leg theorem states a² = c · p and b² = c · q. The altitude theorem states h² = p · q. All three hold only in right-angled triangles.
Exercise 1
A right-angled triangle has c = 10 cm and p = 4 cm. Calculate the leg a.
Show solution and working
a ≈ 6.32 cm
Leg theorem: a² = c · p = 10 cm · 4 cm = 40 cm². So a = √40 cm ≈ 6.32 cm.
Exercise 2
A right-angled triangle has p = 4 cm and q = 9 cm. Calculate the altitude h.
Show solution and working
h = 6 cm
Altitude theorem: h² = p · q = 4 cm · 9 cm = 36 cm². So h = 6 cm.
Exercise 3
A right-angled triangle has c = 10 cm and p = 4 cm. How large is q?
Show solution and working
6 cm
The hypotenuse is made up of the two sections: q = c − p = 10 cm − 4 cm = 6 cm.
Exercise 4
Does the altitude theorem also hold in a triangle without a right angle?
Show solution and working
No
The leg and altitude theorems require a right angle. Without one these relationships do not hold.
Common mistake
Swapping p and q: p belongs to leg a and q to leg b. And applying the theorems to triangles without a right angle.
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