Question

Draw and label all three sides of a right triangle that has 40 degrees angle and a hypotenuse of 10cm

Answer 1

Consider a right angle triangle ABC such that B =90 degree as shown in figure given below with all the three sides labelled.

let
`\angle C = 40^(\circ)`

Sum of all the measures of an angle in a triangle is 180 degree.

In triangle ABC

`\angle A +\angle B +\angle C =180^(\circ)`

`40^(\circ)+90^(\circ)+\angle C= 180^(\circ)`

or

`130^(\circ)+\angle C= 180^(\circ)`

Simplify:

`\angle C= 50^(\circ)`

The side AC (Hypotenuse) = 10 cm

Now, find the other sides i.e, AB and BC;

Using Sine rule

In a right triangle, the sine of an angle is the length of the opposite side divided by the length of the hypotenuse side.

then

`\sin B = (opposite side)/(Hypotenuse side) =(AB)/(AC)`

or

`\sin 40^(\circ) = (AB)/(10)`

or

`AB =10 * \sin 40^(\circ)`

Simplify:

AB = 6.4278761 ≈ 6.43 cm

Now,by using tangent rule to solve for BC

In a right triangle, the tangent of an angle is the length of the opposite side divided by the length of the adjacent side.

then;

`\tan B = (opposite side)/(Adjacent side) =(AB)/(BC)`

or

`\tan 40^(\circ) = (6.4278761)/(BC)`

or

`BC =(6.4278761)/(\tan 40^(\circ))`

Simplify:

BC = 7.66044443≈ 7.66 cm

therefore, the sides AB and BC of the triangle ABC are 6.43(approx) and 7.66(approx.)