In the diagram below, is an isosceles triangle with PQ =PR. QPR = 54° and QR= 20cm calculate to 2s.f
A PQR
B. The side PR
C The Altitude of the triangle
D. The sum of PQR
Since PQ = PR, the triangle PQR is isosceles and therefore, angle QPR = angle QRP.
Now we can calculate angle RPQ:
Angle QPR + angle QRP + angle RPQ = 180°
54° + 54° + angle RPQ = 180°
108° + angle RPQ = 180°
angle RPQ = 180° - 108°
angle RPQ = 72°
Now, we can use the sine rule to calculate PR:
sin(54°)/20 = sin(72°)/PR
PR = 20 * sin(72°) / sin(54°)
PR ≈ 24.66 cm
To calculate the altitude of the triangle (PH):
Since PQR is an isosceles triangle, the altitude from P will bisect the base QR, intersecting QR at point H.
Using trigonometry in right triangle PHR:
tan(54°) = PH / HR
PH = HR * tan(54°)
PH ≈ 20 * tan(54°)
PH ≈ 31.95 cm
The sum of angles in a triangle is always 180°, so the sum of triangle PQR is:
PQR + QPR + QRP = 180°
54° + 54° + 72° = 180°
180° = 180°
Therefore:
A. PQR is an isosceles triangle
B. PR ≈ 24.66 cm
C. The altitude PH ≈ 31.95 cm
D. The sum of PQR = 180°