Circles Γ1 and Γ2 intersect at 2 distinct points A and B. A line l through A intersects Γ1 and Γ2 at C and D, respectively. Let M be the midpoint of CD. The line MB intersects Γ1 and Γ2 again at E and F, respectively. If MA=129,MB=156 and MC=182, what is the value of EF?
I worked this earlier:
29*182 = ME*156, so ME = 33.8333
MD = MC = 129+182 = 311
MF*156 = 129*311, so MF = 257.173
EF = MF+ME = 33.833+257.173 = 291
Is my answer incorrect?
Incorrect.
The correct answer is 301.
Steve, in the first line is 129, not 29... and you shouldn't work with decimals because in the end the ugly dominator simplifies with the numerator.
To solve this problem, we need to apply the properties of intersecting circles and use the given information to find the value of EF.
Step 1: Draw a diagram
Start by drawing circles Γ1 and Γ2 intersecting at two distinct points, A and B. Then draw a line l through point A intersecting the circles at points C and D, respectively. Label the midpoint of CD as M. Finally, draw a line MB intersecting the circles again at points E and F.
Step 2: Use the properties of intersecting circles
Since line l intersects circles Γ1 and Γ2, we can use the property that the angles formed by intersecting chords are equal. Therefore, angles ACB and ADB, as well as angles ACF and ADF, are equal.
Step 3: Use the properties of a midpoint
As M is the midpoint of CD, we know that MC = MD. This implies that triangles MCA and MDB are congruent by the Side-Angle-Side (SAS) congruence condition.
Step 4: Find the length of CM
Since MC = MD and triangles MCA and MDB are congruent, we can say that angle MCA = angle MDB. We also know that angle ACB = angle ADB by the property of intersecting chords. Therefore, triangles MCA and MDB are similar due to angle-angle similarity condition.
By the similarity of triangles MCA and MDB, we have:
MC/MA = MD/MB
Substituting the given values:
MC/129 = MC/156
Cross-multiplying and solving for MC:
MC * 156 = 129 * MC
156 = 129
Therefore, MC = 182.
Step 5: Apply the properties to find EF
Since triangles MCA and MDB are congruent, we have CD = CA + AD = 2 * MC = 2 * 182 = 364.
Using the properties of the intersecting chords, we know that:
AC * AD = BC * BD
Substituting the values:
CA * AD = CB * BD
Since CA = CD - DA and CB = CD - DB:
(CD - DA) * AD = (CD - DB) * BD
Expanding the equation:
CD * AD - DA * AD = CD * BD - DB * BD
Since DA = DB and CD = 364:
(364 - DA) * AD = 364 * BD - DB^2
Substituting given values:
(364 - DA) * AD = 364 * BD - DB^2
Since AD = MC = 182, we have:
(364 - DA) * 182 = 364 * BD - DB^2
Simplifying the equation:
364^2 - 182 * DA = 364 * BD - DB^2
From the given information, we know that DA = 129, DB = MC = 182:
364^2 - 182 * 129 = 364 * BD - 182^2
Simplifying further:
364^2 - 364 * 182 = 364 * BD - 182^2
364 * (364 - 182) = 364 * BD - 182^2
364 * 182 = 364 * BD - 182^2
Simplifying:
364 * BD = 364 * 182 + 182^2
BD = (364 * 182 + 182^2) / 364
BD = 182 * (364 + 182) / 364
BD = (182 * 546) / 364
BD = 273
Since EF is part of the same chord as BD, EF will also have the same length as BD.
Therefore, EF = BD = 273.