Question

The triangle ABC pictured below is a right, isosceles triangle. If the length of side AC is 3, give the lengths of the other two sides and the measures of angle A and angle B.

Answer 1

Explanation

From the statement, we know that we have a right triangle that:

0. has an angle ∠C = 90°,

,

1. is also an isosceles triangle,

,

2. has a side AC = 3.

1) From points 2 and 3, we know that:

$AC=BC=3.$

Because we have a right triangle, we can use Pitagoras Theorem, which states that:

$c^2=a^2+b^2.$

Where:

• c = AB = hypotenuse,

,

• a = BC = 3,

,

• b = AC = 3.

Replacing these data in the equation above, we get:

$c^2=3^2+3^2=9+9=18\Rightarrow AB=c=\sqrt{18}.$

2) From point 2 we know that angles A and B must be equal:

$\angle A=\angle B.$

From geometry, we know that the inner angles of a triangle sum up to 180°, so we have:

$\angle A+\angle B+\angle C=180\degree\Rightarrow2\angle A+\angle C=180\degree\Rightarrow\angle A=\frac{180\degree-\angle C}{2}=\frac{180\degree-90\degree}{2}=45\degree.$

Where we have used point 1.

Answer

The sides of the triangle are:

• AB = √18,,

,

• AC = 3,

,

• BC = 3.

The angles of the triangle are:

• ∠A = 45°,,

,

• ∠B = 45°,,

,

• ∠C = 90°.