PQR is an isosceles triangle in which PQ =PR if QR =16cm and the height PM =15cm,calculate PQ
Since triangle PQR is isosceles with PQ = PR, we can conclude that angle PQR = angle PRQ. This means that triangle PQR is also an equilateral triangle.
Since PM is the height of triangle PQR, we can use the height to calculate any side of the equilateral triangle. In an equilateral triangle, the height is also the perpendicular bisector of the base.
Let's call the midpoint of QR as N. So, PN = NM = 8 cm (half of QR)
Now, let's draw line segment PN.
Since triangle PNM is a right-angled triangle with PM as the height, we can use Pythagoras theorem to find PN.
Using Pythagoras theorem:
PN^2 = PM^2 - NM^2
PN^2 = 15^2 - 8^2
PN^2 = 225 - 64
PN^2 = 161
PN ≈ √161 ≈ 12.689 cm
Since triangle PQR is equilateral, PQ = QR = PR.
Therefore, PQ ≈ 12.689 cm.
To calculate the length of PQ in an isosceles triangle, we can use the Pythagorean theorem.
Step 1: Draw a diagram of the triangle.
Let's label the triangle as follows:
PQ = PR (isosceles)
QR = 16 cm
PM = 15 cm (height)
Q
/ \
/ \
P ------ R
Step 2: Identifying the right triangle.
We can see that triangle PQM is a right triangle because the height PM is perpendicular to the base QR.
Step 3: Using the Pythagorean theorem.
In a right triangle, the square of the hypotenuse (PQ) is equal to the sum of the squares of the other two sides (PM and MQ).
PQ^2 = PM^2 + MQ^2
PQ^2 = 15^2 + (QR/2)^2 (since PQ=PR, so MQ=QR/2)
Step 4: Substitute the given values.
PQ^2 = 15^2 + (16/2)^2
PQ^2 = 225 + 8^2
PQ^2 = 225 + 64
PQ^2 = 289
Step 5: Take the square root to find PQ.
PQ = √289
PQ = 17 cm
Therefore, the length of PQ is 17 cm.
To solve this problem, we can use the Pythagorean theorem and the property of isosceles triangles.
First, let's draw the triangle PQR with the given information:
P
/ \
/ \
/ \
/ \
/ \
Q-----------R
Since PQR is an isosceles triangle, we know that PQ = PR. Let's denote this length as x.
To find the length of PQ, we can consider the right-angled triangle PMQ, where PM is the height and QM is the base.
Using the Pythagorean theorem, we have:
PM^2 + QM^2 = PQ^2
Substituting the given values, we have:
15^2 + QM^2 = x^2
225 + QM^2 = x^2
Next, we need to find the length of QM. Since QM is the base of the right-angled triangle PMQ, we can calculate it using the Pythagorean theorem as well.
Using the given information, we have:
QR^2 = QM^2 + MR^2
Substituting the given values, we have:
16^2 = QM^2 + x^2
256 = QM^2 + x^2
Now, we have two equations:
225 + QM^2 = x^2 (equation 1)
256 = QM^2 + x^2 (equation 2)
Since both equations have QM^2 + x^2 on the right side, we can equate them and solve for x:
225 + QM^2 = 256
QM^2 = 256 - 225
QM^2 = 31
Now, substituting QM^2 = 31 into equation 1:
225 + 31 = x^2
256 = x^2
Taking the square root of both sides:
x = √256
x = 16
Therefore, PQ = PR = 16 cm.