Question

Find the area and perimeter of ABC at right. Give approximate (decimal) answers, not exact answers

Answer 1

Area of Δ ABC = 21.86 units square

Perimeter of Δ ABC = 24.59 units

Explanation:

Given:

In Δ ABC

∠A=45°

∠C=30°

Height of triangle = 4 units.

To find area and perimeter of triangle we need to find the sides of the triangle.

Naming the end point of altitude as 'D'

Given
`BD\perp AC`

For Δ ABD

Since its a right triangle with one angle 45°, it means it is a special 45-45-90 triangle.

The sides of 45-45-90 triangle is given as:

Leg1
`=x`

Leg2
`=x`

Hypotenuse
`=x\sqrt2`

where
`x` is any positive number

We are given BD(Leg 1)=4

∴ AD(Leg2)=4

∴ AB (hypotenuse)
`=4\sqrt2=5.66`

For Δ CBD

Since its a right triangle with one angle 30°, it means it is a special 30-60-90 triangle.

The sides of 30-60-90 triangle is given as:

Leg1(side opposite 30° angle)
`=x`

Leg2(side opposite 60° angle)
`=x\sqrt3`

Hypotenuse
`=2x`

where
`x` is any positive number

We are given BD(Leg 1)=4

∴ CD(Leg2)
`=4\sqrt3=6.93`

∴ BC (hypotenuse)
`=2* 4=8`

Length of side AC is given as sum of segments AD and CD

`AC=AD+CD=4+6.93=10.93`

Perimeter of Δ ABC= Sum of sides of triangle

⇒ AB+BC+AC

⇒
`5.66+8+10.93`

⇒
`24.59` units

Area of Δ ABC =
`(1)/(2)* base* height`

⇒
`(1)/(2)* 10.93* 4`

⇒
`21.86` units square