PQR ~ STU. PQ = 12cm and ST = 9cm. The area of STU is 72 cm². Find the area of PQR.
The ratio of their areas is proportional to the ratio of the
sides squared.
A(pqr) = (12/9)^2 * 72 = 128 sq.cm.
Well, first things first, let me just say that PQR and STU sound like they could use some better names. How about we call them Bob and Fred? Much more relatable.
Now, let's get down to business. We have the sides PQ and ST, and we know that PQ equals 12 centimeters while ST equals 9 centimeters. And lucky for us, we also have the area of STU, which is 72 square centimeters.
To find the area of PQR, we need to find the missing side QR. We can do that by using the formula for the area of a triangle: Area = (base * height) / 2.
Since we already know the area (72 cm²) and the base (ST), we can rearrange the formula to solve for the height. Multiply both sides by 2, divide by the base and voila! We have the height.
But wait, PQR and STU are similar triangles, which means their corresponding sides are proportional. So, if the ratio of QR to TU is the same as the ratio of PQ to ST, we're golden!
Since PQ is 12 cm and ST is 9 cm, their ratio is 12/9 = 4/3. And since QR is our missing side and TU is the base of the larger triangle, we can set up a proportion: QR/TU = 4/3.
Now, we know that the area of a triangle is equal to (base * height) / 2. Plugging in our values, we get the equation (9 * QR) / 2 = 72.
Solving for QR, we multiply both sides by 2 and divide by 9. Good old math, always keeping us on our toes. QR equals... *drumroll*... 16 centimeters!
So now, using our newfound knowledge of QR, we can calculate the area of PQR using the same formula: (base * height) / 2. Plugging in PQ as the base (12 centimeters) and QR as the height (16 centimeters), we get (12 * 16) / 2 = 192 / 2 = 96 square centimeters.
So, the area of PQR, or Bob, is 96 square centimeters.
To find the area of triangle PQR, we can use the ratio of the sides to determine the ratio of the areas. Since PQR ~ STU, we know that the ratio of their corresponding side lengths is equal.
Let's represent the ratio of the side lengths as x:
PQ/ST = QR/TU = PR/SU = x
Given that PQ = 12 cm and ST = 9 cm, we can set up the following equation:
12/9 = x
Simplifying the equation, we get:
4/3 = x
Now, let's use the ratio of the side lengths to find the ratio of the areas:
Area(PQR)/Area(STU) = (PQ/ST)^2 = (12/9)^2 = (4/3)^2 = 16/9
We know that the area of STU is 72 cm², so we can set up the following equation:
Area(PQR)/72 = 16/9
To find the area of PQR, we can cross-multiply and solve for it:
Area(PQR) = (16/9) * 72 = 128 cm²
Therefore, the area of triangle PQR is 128 cm².
To find the area of PQR, we can use the concept of similar triangles. Since PQR is similar to STU, the ratio of their corresponding side lengths is the same.
Given: PQR ~ STU, PQ = 12cm, ST = 9cm, and the area of STU is 72 cm².
Step 1: Find the ratio of the corresponding side lengths.
Since PQR is similar to STU, we can set up a ratio of the corresponding side lengths:
PQ/ST = QR/TU = PR/SU
In this case, we know that PQ = 12cm and ST = 9cm, so the ratio becomes:
12/9 = QR/TU = PR/SU
Simplifying the ratio, we have:
4/3 = QR/TU = PR/SU
Step 2: Find the length of QR.
Using the ratio above, we know that the length of QR is 4/3 times the length of TU.
Since TU is equal to ST, which is 9cm, we can substitute the values and find the length of QR:
QR = (4/3) * 9cm
QR = 36/3
QR = 12 cm
Step 3: Find the area of PQR.
The area of a triangle can be calculated using the formula:
Area = (base * height) / 2
In this case, PQ is the base and QR is the height. Substituting the values, we have:
Area of PQR = (12cm * 12cm) / 2
Area of PQR = 144 / 2
Area of PQR = 72 cm²
Therefore, the area of PQR is 72 cm².