ABCD is a parallelogram. E is a point on AB such that 234×AE=EB. Let DE intersect AC at F. What is the ratio AC:AF?
oin AC and DE. Triangles AFE and DFC are similar.
so AF/FC = AE/CD = AE/EB also AE + EB = AB ie AE +234AE =AB so AE/AB =1/235
so AF/FC = 1/235 FC = 235 AF now, AC = AF +FC = 236AF ie AC/AF = 236/1
ie AC : AF = 236 : 1
To find the ratio AC:AF, we need to determine the value of AF in terms of AC. Let's proceed step by step.
Given: ABCD is a parallelogram and E is a point on AB such that 234×AE = EB.
Step 1: Determine the relationship of AE and AB.
Since E is a point on AB, we can say that AE : EB = AB : AE + EB.
Substituting the given values: AE : EB = AB : AE + 234×AE.
Step 2: Simplify the equation obtained in Step 1.
Multiply both sides of the equation by AE to eliminate the ratio:
AE × AE : AE × EB = AB × AE: AE × AE + 234×AE × AE.
Simplifying further: AE^2 : AE × EB = AB × AE: AE^2 + 234×AE^2.
Step 3: Use the property of a parallelogram.
In a parallelogram, opposite sides are equal in length.
Since ABCD is a parallelogram, AB = CD, and AE = DC in terms of length.
Step 4: Substitute the values obtained in Step 3 into the equation obtained in Step 2.
AE^2 : AE × EB = AE × EB: AE^2 + 234×AE^2.
Simplifying further: AE^2 : AE × EB = AE × EB: AE^2 + 234×AE^2.
Step 5: Simplify the equation.
Cancel out the common factor AE from both fractions:
AE : EB = 1: 1 + 234.
Reducing further: AE : EB = 1: 235.
Step 6: Determine the relationship of AF and AC.
In triangle ADC, DE is parallel to BC, so by similarity of triangles ADC and BCF, we can say that AC: AF = CD : EB.
Step 7: Substitute the values for CD and EB.
Since CD = AB and AE: EB = 1: 235, we have:
AC: AF = AB : EB = AE : EB = 1: 235.
Therefore, the ratio AC : AF is 1: 235.