Question

find the unknown side length.

solve.

a diagonal divides a square cake that is 8 in. on a side. what is the area on either side of the diagonal?

the math club has a flag in the shape of a right triangle. it has a height of 12 in. and a base of 27 in. what is the area of the flag?

Explanation:

Step1: Use the formula for the area of a triangle \(A=\frac{1}{2}bh\)

For problem 1: Given \(A = 30\space in^{2}\), \(b=6\space in\). Rearrange the formula to \(h=\frac{2A}{b}\). Substitute \(A = 30\) and \(b = 6\): \(h=\frac{2\times30}{6}\).

Step2: Calculate the value

\(h=\frac{60}{6}=10\space in\)

For problem 2: Given \(A = 12\space cm^{2}\), \(h = 6\space cm\). Rearrange the formula to \(b=\frac{2A}{h}\). Substitute \(A = 12\) and \(h = 6\): \(b=\frac{2\times12}{6}\).

Step3: Calculate the value

\(b=\frac{24}{6}=4\space cm\)

For problem 3: Given \(A = 18\space m^{2}\), \(b = 12\space m\). Rearrange the formula to \(h=\frac{2A}{b}\). Substitute \(A = 18\) and \(b = 12\): \(h=\frac{2\times18}{12}\).

Step4: Calculate the value

\(h=\frac{36}{12}=3\space m\)

For problem 4: Area of the square \(A_{square}=s^{2}\), \(s = 8\space in\), so \(A_{square}=8^{2}=64\space in^{2}\). The diagonal divides the square into two equal - area triangles. So the area on either side \(A=\frac{1}{2}\times64 = 32\space in^{2}\)

For problem 5: Use the formula \(A=\frac{1}{2}bh\). Given \(b = 27\space in\), \(h = 12\space in\). Substitute into the formula: \(A=\frac{1}{2}\times27\times12\). First, \(27\times12 = 324\), then \(A=\frac{324}{2}=162\space in^{2}\)

Answer:

1. \(h = 10\space in\)
2. \(b = 4\space cm\)
3. \(h = 3\space m\)
4. \(32\space in^{2}\)
5. \(162\space in^{2}\)