Q34.If two intersecting chords of a circle make equal angles with the diameter passing through their point of intersection, prove that the chords are equal.
To prove that two intersecting chords of a circle are equal when they make equal angles with the diameter passing through their point of intersection, we can use the following steps:
Let's consider a circle with center O, diameter AB, and two intersecting chords CD and EF.
Step 1: Draw the diameter passing through the point of intersection of the chords (denoted as P).
Step 2: Since CD and EF make equal angles with the diameter passing through P, we can conclude that ∠CPD = ∠EPF.
Step 3: By the inscribed angle theorem, the measure of an angle formed by an arc is half the measure of its intercepted arc. Therefore, ∠CPD intercepts arc CD and ∠EPF intercepts arc EF.
Step 4: Since ∠CPD = ∠EPF and ∠CPD intercepts arc CD while ∠EPF intercepts arc EF, we can conclude that arc CD is congruent to arc EF.
Step 5: Congruent arcs in a circle correspond to equal chords. Therefore, we can conclude that CD is equal to EF.
Hence, we have proven that if two intersecting chords of a circle make equal angles with the diameter passing through their point of intersection, then the chords are equal.
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To prove that two intersecting chords of a circle are equal when they make equal angles with the diameter passing through their point of intersection, we can use the following steps:
Let's consider a circle with center O, diameter AB, and two intersecting chords CD and EF.
Step 1: Draw the diameter passing through the point of intersection of the chords (denoted as P).
Step 2: Since CD and EF make equal angles with the diameter passing through P, we can conclude that ∠CPD = ∠EPF.
Step 3: By the inscribed angle theorem, the measure of an angle formed by an arc is half the measure of its intercepted arc. Therefore, ∠CPD intercepts arc CD and ∠EPF intercepts arc EF.
Step 4: Since ∠CPD = ∠EPF and ∠CPD intercepts arc CD while ∠EPF intercepts arc EF, we can conclude that arc CD is congruent to arc EF.
Step 5: Congruent arcs in a circle correspond to equal chords. Therefore, we can conclude that CD is equal to EF.
Hence, we have proven that if two intersecting chords of a circle make equal angles with the diameter passing through their point of intersection, then the chords are equal.
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