Perimeter of a Triangle: Using Pythagoras' theorem

To solve the problem of finding the perimeter of triangle ABD, we will apply the following steps:

• Step 1: Identify the given dimensions of the rectangle.
• Step 2: Calculate the length of the diagonal BD using the Pythagorean theorem.
• Step 3: Sum the sides of triangle ABD to find its perimeter.

Now, let's work through each step:

Step 1: We know from the problem that AB = 15 and AD = 8.

Step 2: The triangle ABD is a right triangle with AB and AD as the legs, and BD as the hypotenuse. Therefore, by the Pythagorean theorem:

Calculating these squares gives:

Taking the square root of both sides, we find:

Step 3: Now, calculate the perimeter of triangle ABD.

Therefore, the perimeter of triangle ABD is $40$.

We calculate the perimeter of the triangle:

$12+4\sqrt{5}=4+AC+BC$

As we want to find the hypotenuse (BC), we isolate it:

$12+4\sqrt{5}-4-AC=BC$

$BC=8+4\sqrt{5}-AC$

Then calculate AC using the Pythagorean theorem:

$AB^2+AC^2=BC^2$

$4^2+AC^2=(8+4\sqrt{5}-AC)^2$

$16+AC^2=(8+4\sqrt{5})^2-2\times AC(8+4\sqrt{5})+AC^2$

We then simplify the two:$AC^2$

$16=8^2+2\times8\times4\sqrt{5}+(4\sqrt{5})^2-2\times8\times AC-2AC4\sqrt{5}$

$16=64+64\sqrt{5}+16\times5-16AC-8\sqrt{5}AC$

$16AC+8\sqrt{5}AC=64+64\sqrt{5}+16\times5-16$

$AC(16+8\sqrt{5})=128+64\sqrt{5}$

$AC=\frac{128+64\sqrt{5}}{16+8\sqrt{5}}=\frac{8(16+8\sqrt{5})}{16+8\sqrt{5}}$

We simplify to obtain:

$AC=8$

Now we can replace AC with the value we found for BC:

$BC=8+4\sqrt{5}-AC$

$BC=8+4\sqrt{5}-8=4\sqrt{5}$

To determine which has the larger measurement, the triangle's perimeter or the circle's circumference, we need to compute both values.

Step 1: Calculate the perimeter of the Triangle
We are given two sides of the triangle: 6 and 5. Since it's implied to be a right triangle, we apply the Pythagorean theorem to find the third side, the hypotenuse $c$:

The perimeter $P$ of the triangle is:

Step 2: Calculate the circumference of the Circle
The circumference $C$ of a circle with radius $r$ is given by the formula:

Assuming the radius of the circle is equivalent to the '6' mentioned for the green line in the SVG:

Step 3: Compare the Triangle's Perimeter and the Circle's Circumference
We compare the values:

• Perimeter of the Triangle: $P \approx 18.81$
• Circumference of the Circle: $C \approx 37.7$

The circumference of the circle (37.7) is greater than the perimeter of the triangle (18.81).

Therefore, the circle has the larger measurement.

Conclusion: The circle has the larger perimeter.