In a triangle ABC,E and F are the mid-points of AC and AB respectively. The altitude AP to BC intersects EF at Q.Prove AQ =PB.
I would use similar triangles
First ACP ~ AFQ (use alt angle of parallel lines, and one common angle, triangles with three same angles are sim)
Then, AF=FC given midpoint.
which then, using proportion, aq=qp
To prove that AQ = PB, we can make use of the concept of triangles and their properties. Let's break down the steps to prove this:
Step 1: Draw the given triangle ABC. Label the midpoints of AC and AB as E and F, respectively.
Step 2: Draw the altitude AP from vertex A to side BC. Label the point of intersection between AP and EF as Q.
Step 3: Now, let's look at triangle ABC. Since E is the midpoint of AC, we can apply the Midpoint Theorem, which states that a line segment connecting the midpoints of two sides of a triangle is parallel to and half the length of the third side. Therefore, we can conclude that EF is parallel to BC and that EF = 0.5 * BC.
Step 4: Since EF is parallel to BC, we can also apply the Alternate Interior Angles Theorem, which states that if a transversal intersects two parallel lines, then the alternate interior angles are congruent. Therefore, angle AEF is congruent to angle ABC.
Step 5: Now, let's look at triangle AEQ. In this triangle, angle AEF and angle A are vertical angles, which means they are congruent. Therefore, angle AEF is congruent to angle A.
Step 6: Since angle AEF is congruent to angle ABC, and angle AEF is also congruent to angle A, we can conclude that angle ABC and angle A are congruent.
Step 7: Now, let's look at triangle APB. In this triangle, angle A is congruent to angle ABC. Also, angle APB is a right angle because it is formed by the altitude AP. Therefore, triangle APB is a right triangle.
Step 8: In a right triangle, the height (AP) and the hypotenuse (AB) are related to each other through the Pythagorean Theorem. According to the Pythagorean Theorem, the square of the length of the height (AP) plus the square of the length of the base (PB) is equal to the square of the length of the hypotenuse (AB).
Step 9: However, we are not given the lengths of the sides of the triangle in this problem. But what we can do is use similar triangles to establish a relationship between AQ and PB.
Step 10: Notice that triangle AEQ is similar to triangle ABC, because angle A is congruent to angle ABC and angle E is congruent to angle B (since E is the midpoint of AC and F is the midpoint of AB). Therefore, we can write the proportion:
AQ / AB = EQ / BC
Step 11: Recall that EF = 0.5 * BC from Step 3. Therefore, EQ is also equal to 0.5 * BC.
AQ / AB = (0.5 * BC) / BC
Step 12: Simplifying, we get:
AQ / AB = 0.5
AQ = 0.5 * AB
AQ = PB
Step 13: Therefore, we have proved that AQ = PB by using the concept of similar triangles and the properties of right triangles.
Note: The exact values of AQ and PB cannot be determined without more information about the lengths of the sides, but we have shown that they are equal based on the given conditions.