Question

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

Answer 1

Answer:

Area of Δ ABC = 21.86 units square

Perimeter of Δ ABC = 24.59 units

Step-by-step 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) $=2\times 4=8$  

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 = $\frac{1}{2}\times base\times height$

⇒  $\frac{1}{2}\times 10.93\times 4$

⇒ $21.86$ units square