Bunuel wrote:
rvinodhini wrote:
Hi
I need a quick clarification on the concept of perpendicular bisector.
With a perpendicular bisector, the bisector always crosses the line segment at right angles
If any line cuts another line at 90 then it should be a perpendicular bisector right - i.e it divided the line segment into equal halves at 90 ?
So here BC should be the perpendicular bisector and the AC=CD=3 right ?
Please let me know what am missing here.
I do understand the explanations in the other thread mentioned,but can someone clarify as to why AC is not the perpendicular bisector ?
A perpendicular bisector is a line which cuts a line segment into two equal parts at 90°. So AC to be a perpendicular bisector of BD it must not only cut it at 90° (which it does) but also cut it into two equal parts. Now, in order AC to cut BD into two equal parts right triangle ABD must be isosceles, which, as it turns out after some math, it is not.
Complete solution:
In triangle ABC, if BC = 3 and AC = 4, then what is the length of segment CD?A. 3
B. 15/4
C. 5
D. 16/3
E. 20/3
Attachment:
splittingtriangle.jpg
Important property:
perpendicular to the hypotenuse will always divide the triangle into two triangles with the same properties as the original triangle.
Thus, the perpendicular AC divides right triangle ABD into two similar triangles ACB and DCA (which are also similar to big triangle ABD). Now, in these three triangles the ratio of the corresponding sides will be equal (corresponding sides are the sides opposite the same angles marked with red and blue on the diagram).
So, \(\frac{CD}{AC}=\frac{AC}{BC}\) --> \(\frac{CD}{4}=\frac{4}{3}\) --> \(CD=\frac{16}{3}\).
Answer: D.
For more on this subject please check Triangles chapter of Math Book:
math-triangles-87197.htmlHope it helps.
This part is clear:
triangles the ratio of the corresponding sides will be equal (corresponding sides are the sides opposite the same angles)However, how does one determine which angles are equal? Except 90 degree angles of both triangles, i could not seem to follow how exactly other angles became equal?
Both smaller triangles are similar to the large triangle. So they are similar to each other too.
In triangles BAD and BCA,
Angle BAD = BCA (90 degrees)
and angle B is common in both
So by AA, triangles BAD and BCA are similar
Similarly, in triangles BAD and ADC,
angle BAD = ACD (90 degrees)
and angle D is common in both
So by AA, triangles BAD and ACD are similar
So triangle BAD is similar to triangle BCA and ACD so triangle BCA is similar to triangle ACD too.
Thanks for explaining this. I understand how 3 triangles are similar to each other. However how do we determine which side is similar to which in order to set up the ratio.
for example, if we take two smaller triangles - Triangle ABC and ADC....I think side AB is corresponding to AD, BC is corresponding to CD and AC is common. Is that correct? if yes, how do I find which side is corresponding in the smaller triangle with respect to the bigger triangle?