Question

If the hypotenuse of a 30°-60°-90° Triangle is 10√2, find the length of the other two sides.

Answer 1

Let us consider a 30°-60°-90°△ABC right angled at B in which ∠C = 30⁰ and ∠A = 60⁰ with hypotenuse AC = 10√2 units.

Solution:-

In △ABC,

$\longrightarrow$sin ∠C = $\sf \dfrac{Perpendicular}{Hypotenuse}$

$\longrightarrow$sin ∠C = $\sf \dfrac{AB}{AC}$

$\longrightarrow$sin 30⁰ = $\sf \dfrac{AB}{10\sqrt{2}}$

$\longrightarrow$$\sf \dfrac{1}{2}= \dfrac{AB}{10\sqrt{2}}$

$\longrightarrow$$\sf AB = \dfrac{10\sqrt{2}}{2}$

$\longrightarrow$$\sf AB = 5\sqrt{2}\:units$

In △ABC,

$\longrightarrow$cos ∠C = $\sf \dfrac{Base}{Hypotenuse}$

$\longrightarrow$cos ∠C = $\sf \dfrac{BC}{AC}$

$\longrightarrow$cos 30⁰ = $\sf \dfrac{BC}{10\sqrt{2}}$

$\longrightarrow$$\sf \dfrac{\sqrt{3}}{2}= \dfrac{BC}{10\sqrt{2}}$

$\longrightarrow$$\sf BC = \dfrac{10\sqrt{2}\times \sqrt{3}}{2}$

$\longrightarrow$$\sf BC = \dfrac{10\sqrt{2\times 3}}{2}$

$\longrightarrow$$\sf BC = \dfrac{10\sqrt{6}}{2}$

$\longrightarrow$$\sf BC = 5\sqrt{6}\: units$

Thus , the length of other two sides of triangles are 5√2 and 5√6 units.