Given triangleABC proportion to trianglePQR. If AB=4cm, BC=6cm, AC=9cm, and QR=4cm, find the perimeter of trianglePQR. What is the ratio of similitude?
since the common sides have a constant ratio,
AB/PQ = BC/QR = CA/SP
4/PQ = 6/4 = 9/SP
the ratio is 6/4 = 3/2
so, the perimeters are also in that ratio, and
P(ABC) = 4+6+9 = 19
P(PQR) = 19 * 3/2
Two supplementary angles are in the ratio 5:7. Find the larger angle
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Supplementary angles sum up to 180 degrees.
Taking the larger one as 7x, the smaller one is 5x (they are in a 5:7 ratio)
=> 5x + 7x = 180
=> 12x = 180
=> x = 15
Hence, the larger angle = 7x = 7(15) = 105
To find the perimeter of triangle PQR, you need to find the lengths of all three sides (PQ, QR, and RP) and then add them together.
Given that triangle ABC is proportional to triangle PQR, the corresponding sides of both triangles are in proportion. So we can set up the following proportion:
AB/PQ = BC/QR = AC/RP
We are given AB = 4cm, BC = 6cm, and QR = 4cm. Let's substitute these values into the proportion:
4/PQ = 6/4 = 9/RP
Simplifying the second fraction, we get:
4/PQ = 3/2 = 9/RP
Now we can solve for PQ and RP:
Cross-multiplying the first equation, we get:
2 * 4 = 3 * PQ
8 = 3PQ
PQ = 8/3 cm
Cross-multiplying the second equation, we get:
9 * 2 = RP * 4
18 = 4RP
RP = 18/4 cm
RP = 9/2 cm
Now, we have the lengths of two sides of triangle PQR:
PQ = 8/3 cm and RP = 9/2 cm
To find the third side QR, we can use the given length of QR = 4cm.
Now, let's calculate the perimeter of triangle PQR:
Perimeter = PQ + QR + RP
Perimeter = 8/3 + 4 + 9/2
To add these fractions, you need a common denominator:
Perimeter = (16 + 24 + 27) / 6
Perimeter = 67 / 6 cm
Therefore, the perimeter of triangle PQR is 67/6 cm.
Now let's find the ratio of similitude between triangle ABC and triangle PQR.
The ratio of similitude is determined by the ratio of corresponding side lengths between the two triangles.
In this case, we compare the lengths of AB and PQ. We have AB = 4cm and PQ = 8/3 cm.
To find the ratio of similitude, divide the length of PQ by AB:
Ratio of similitude = PQ / AB
Ratio of similitude = (8/3) / 4
Ratio of similitude = 8/12
Ratio of similitude = 2/3
Therefore, the ratio of similitude between triangle ABC and triangle PQR is 2/3.