A triangles shows the altitude qt and pm of triangle pqr. If qr=8cm , pr=7cm and qt=4cm what is pm?
Given triangle p, q and r.
Given altitude qt to pr = 4.
Shown altitude pm to qr.
qr = 8, pr = 7 and qt = 4.
Triangle area = 7(4)/2 = 14.
Triangle area = 8(qm)/2 = 14 or pm = 3.5
How did you arrive at 3.5
Still not getting the 3.5
To find the length of the segment PM, we can use the concept of similar triangles. The triangles PQR and PTQ are similar because they share angle T and the right angle at Q.
Given that QT is an altitude, it forms a right angle with QR. Therefore, triangle PQT is a right triangle.
Using the similar triangles PQR and PTQ, we can set up the following proportion:
PR / PT = QR / QT
Substituting the given values:
7cm / PT = 8cm / 4cm
Cross-multiplying:
7cm * 4cm = 8cm * PT
28cm = 8cm * PT
Dividing both sides by 8cm:
PT = 28cm / 8cm
PT = 3.5cm
Now, we can use the Pythagorean theorem to find the length of the segment PM:
PM^2 = PT^2 + MT^2
Since PT is 3.5cm and QT is 4cm, MT (the length of the altitude from P to QR) can be found by subtracting QT from PT:
MT = PT - QT
MT = 3.5cm - 4cm
MT = -0.5cm or 0.5cm (taking the positive value)
Now we can substitute the values into the Pythagorean theorem equation:
PM^2 = (3.5cm)^2 + (0.5cm)^2
PM^2 = 12.25cm^2 + 0.25cm^2
PM^2 = 12.5cm^2
Taking the square root of both sides:
PM = √(12.5cm^2)
PM ≈ 3.54cm
Therefore, the length of segment PM is approximately 3.54cm.