Given that triangles PQR~STU:triangle #1 sides 8,9,10 triangle #2 3,?,?
a) What is the ratio of similitude of PQR~STU triangles?
Express answer in fractional form.
b) What is the ratio of the perimeter of PQR~STU triangles?
Express answer in fractional form.
a) the shortest sides are in the ratio 8/3
b) so are the perimeters.
To find the ratio of similitude between two similar triangles, we need to compare their corresponding sides. Let's refer to triangle #1 as PQR and triangle #2 as STU.
a) To find the ratio of similitude, we need to calculate the ratio of corresponding sides. In this case, we can find the ratio by dividing the lengths of corresponding sides. The corresponding sides in this case are PQ and ST, QR and TU, and PR and SU.
Triangle #1 (PQR) has sides of lengths 8, 9, and 10. Triangle #2 (STU) has a known side length of 3, and we need to find the lengths of the other two sides.
We can find the missing side lengths of triangle #2 (STU) using the concept of ratios and proportions. Since the two triangles are similar, the ratio of corresponding side lengths should be the same for all sides.
Using a proportion, we can set up the following equation:
8/3 = 9/x
Cross-multiplying, we get:
8x = 3 * 9
8x = 27
Dividing both sides by 8, we find that x = 27/8.
Therefore, the lengths of the sides in triangle #2 (STU) are 3, 27/8 (or 3.375), and 9.
Now, let's calculate the ratio of similitude:
PQ/PQ = 8/3 = 8:3
QR/TU = 9/3.375 = 36/13.5 = 8/3
PR/SU = 10/9 = 10:9
The ratios for each corresponding side length are 8:3, 8:3, and 10:9 respectively. Hence, the ratio of similitude for triangles PQR~STU is 8:3.
b) To find the ratio of the perimeter of two similar triangles, we can simply divide the perimeters of the triangles. In this case, we need to calculate the perimeter of triangle #1 (PQR) and triangle #2 (STU).
The perimeter of PQR is the sum of its side lengths: 8 + 9 + 10 = 27.
The perimeter of STU can be calculated by adding the side lengths: 3 + 27/8 + 9 = 3 + 27/8 + 72/8 = 99/8.
Now, let's find the ratio of the perimeters:
Perimeter of PQR / Perimeter of STU = 27 / (99/8)
To divide by a fraction, we multiply by its reciprocal:
27 / (99/8) = 27 * (8/99)
Now we can simplify the expression:
27 * 8 = 216
99
So, the ratio of the perimeters of PQR~STU is 216:99, or simplifying it by dividing both values by 9, we get 24:11.
Therefore, the ratio of the perimeter of triangles PQR~STU is 24:11.