In triangle ABC, angle A is 71 degrees. The perpendiculars from vertices B and C to the opposite sides intersect at point P. Find the measure of angle BPC.

In a right triangle, if two perpendiculars from vertices B and C intersect at point P, the angle BPC is 142 degrees. Because the sum of BPC, BAP, and CAP is 180 degrees, we find that angle BPC = 180 - 19 - 19 = 142 degrees.

Explanation:

In triangle ABC with angle A = 71 degrees, when perpendiculars are drawn from vertices B and C to the opposite side intersecting at point P, the angle BPC is desired. This formation is known as an orthocentre in geometry. The orthocenter has a notable property: the sum of angles BPC, BAP, and CAP is 180 degrees.

To find angle BPC, we first calculate the angles BAP and CAP. Since the triangles ABP and ACP are right triangles (as angles at P are 90 degrees), we use the fact that the sum of angles in a triangle is 180 degrees. Hence, angle BAP = 180 - 90 - 71 = 19 degrees and angle CAP = 180 - 90 - 71 = 19 degrees.

Then we use the property of the orthocentre (BPC, BAP, and CAP sum to 180), we find that angle BPC = 180 - 19 - 19 = 142 degrees.

Learn more about Orthocentre here:

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