Point P on the side AB od triangle ABC splits AB so that AP:BP=1:3. If M is the midpoint of CP, then the ratio of the area of triangle BPM to the area of triangle ABC is
A)1/4
B)5/8
C)3/8
D)3/16
I know that the rule is if two triangles have cogruent sltitudes , then the ratio of their area is equal to the ratio of the sides to which those altitudes are draw,
so based on that therom i got C but when i went back to check on my work i got lost on why does this work
it souds so confusing it would be a GREAT help for me if you could explain thanks
I am sure you made a sketch
let triangel APC = x
then triangle PBC = 3x
the whole triangle ABC = 4x
Since M is the midpoint, triangle BPM = (1/2) of 3x
= 3x/2
ratio of ∆BPM : ∆ABC = 3x/2 : 4x
= 3/2 : 4
= 3 : 8
you had C, so correct
thanks
To find the ratio of the area of triangle BPM to the area of triangle ABC, we need to find the lengths of the sides BP and AB.
Given that AP:BP = 1:3, we can assign a value of x to AP and 3x to BP.
Therefore, AB = AP + BP = x + 3x = 4x.
Since M is the midpoint of CP, we can find PM by dividing CP in half.
Since AP:BP = 1:3, CP is divided into 4 equal parts.
Therefore, PM = CP/2 = (AB + BP)/2 = (4x + 3x)/2 = 7x/2.
Now, let's consider the areas of triangles BPM and ABC.
The area of triangle BPM = (1/2) * BP * PM = (1/2) * 3x * (7x/2) = (21/4) * x^2.
The area of triangle ABC = (1/2) * AB * CP = (1/2) * 4x * 4x = 8 * x^2.
Now, we can find the ratio of the two areas.
Ratio = Area of BPM / Area of ABC = [(21/4) * x^2] / [8 * x^2] = (21/4) / 8 = 21/32.
Therefore, the ratio of the area of triangle BPM to the area of triangle ABC is 21/32.
However, none of the options provided match this ratio. So, it seems there might be an error in the question or the given options. Please double-check the question or provide more information if possible.
To solve this problem, let's break it down step by step.
Step 1: Draw the triangle ABC and point P on side AB such that AP:BP=1:3.
Step 2: Draw CM, where M is the midpoint of CP.
Step 3: To find the ratio of the area of triangle BPM to the area of triangle ABC, we need to know the length of the bases and heights of these triangles.
Step 4: Let's label the lengths of the sides of triangle ABC: AB=x, BC=y, and AC=z.
Step 5: Since AP:BP=1:3, we can express the lengths of AP and BP as follows: AP=x/4 and BP=(3/4)x.
Step 6: Now, let's find the lengths of BM and PM. Since M is the midpoint of CP, BP=BM and PM=PC/2.
Step 7: By substituting the values we found earlier, BM=(3/4)x and PM=(1/2)CM.
Step 8: We can determine the length of CM using the Pythagorean theorem. Since triangle ABC is a right triangle, we have AC^2=AB^2+BC^2. Substituting the values, we get z^2=x^2+y^2.
Step 9: Now, we can determine CM using the midpoint theorem. The midpoint theorem states that for any triangle, the line segment connecting the midpoints of two sides is parallel to the third side and half its length. Therefore, CM=(1/2)AP.
Step 10: Plugging in the value of AP, we get CM=(1/2)(x/4)=(1/8)x.
Step 11: Now that we know the lengths of BM and CM, we can find the ratio of the area of triangle BPM to the area of triangle ABC.
Step 12: The height of triangle BPM is PM, and the height of triangle ABC is AC.
Step 13: By using the fact that the ratio of the areas of two triangles is equal to the ratio of their heights, we have:
Area of triangle BPM/Area of triangle ABC = PM/AC = (1/2)CM/AC = (1/2)((1/8)x)/z = (1/16)(x/z).
Step 14: Using the Pythagorean theorem from Step 8, we can express z as z=sqrt(x^2+y^2).
Step 15: Substituting the value of z, we have:
Area of triangle BPM/Area of triangle ABC = (1/16)(x/sqrt(x^2+y^2)).
Step 16: Simplifying further, we can multiply the fraction by sqrt(x^2+y^2)/sqrt(x^2+y^2) to get:
Area of triangle BPM/Area of triangle ABC = (x/sqrt(x^2+y^2))(1/16).
Step 17: Finally, we can conclude that the ratio of the area of triangle BPM to the area of triangle ABC is x/(16*sqrt(x^2+y^2)).
At this point, we cannot determine the specific value of the ratio without knowing the values of x and y. So, without additional information, we cannot select an answer choice.