AP and CQ are perpendicular to BC and AB respectively. If AP : CQ = 3 : 4, the the numerical value of sin A/ sin C.
since β B is common to βs APB and CQB, and both are right triangles, then they are similar. That is,
AP/CQ = AB/BC = 3/4
Now using the law of sines,
sinA/sinC = BC/AB = 4/3
Well, let's try to solve this using a little humor, shall we?
Since AP and CQ are perpendicular to BC and AB respectively, it seems like we have a right triangle situation going on here. Turns out, triangles can be quite "righteous" when it comes to math!
Now, let's use some trigonometry to find the numerical value of sin A / sin C.
First, we can label the angle opposite to AP as angle A and the angle opposite to CQ as angle C. Could they stand for "Angle Amusing" and "Angle Comical"? Who knows, math has its quirks!
Since AP : CQ = 3 : 4, let's assume that AP is 3x and CQ is 4x. Let's call BP = x.
Now, we can use the definition of sine in a right triangle:
sin A = AP / BC = (3x) / (4x) = 3/4
sin C = CQ / BC = (4x) / (4x) = 4/4 = 1
So, the numerical value of sin A / sin C is (3/4) / 1 = 3/4.
Voila! The answer is 3/4. Keep spreading the laughter in your math journey!
To find the value of sin A / sin C, we need to use the given information that AP : CQ = 3 : 4.
Let's assume that AP = 3x and CQ = 4x, where x is a positive constant.
Since AP and CQ are perpendicular to BC and AB respectively, we can see that triangle APC and triangle CQB are similar triangles.
Using the properties of similar triangles, we know that the corresponding sides of similar triangles are proportional.
Therefore, we can set up the following proportion:
AP / AC = CQ / CB
Substituting the values of AP and CQ, we get:
3x / AC = 4x / CB
Cross multiplying, we get:
3x * CB = 4x * AC
Dividing both sides by x, we get:
3CB = 4AC
Now, let's focus on the ratios of the sides of triangle ABC.
Using the Law of Sines, we know that in any triangle ABC:
sin A / AB = sin C / CB
Substituting the values we have found:
sin A / 4x = sin C / 3x
Cross multiplying, we get:
sin A * 3x = sin C * 4x
Dividing both sides by x, we get:
sin A * 3 = sin C * 4
Finally, we can rearrange the equation to find the value of sin A / sin C:
sin A / sin C = 4 / 3
Therefore, the numerical value of sin A / sin C is 4/3.
To find the numerical value of sin A/sin C, we need to find the values of sin A and sin C separately.
First, let's consider triangle ABC. Since AP is perpendicular to BC, triangle ABC and triangle APB are similar triangles by the AA similarity criterion.
Using the similarity, we can set up a proportion:
AB/AP = BC/BP
Since AP : CQ = 3 : 4, we can also set up the proportion:
AP/CQ = 3/4
Rearranging the proportions, we get:
AP = (3/4) * CQ
Now, let's consider triangle ACQ. Since CQ is perpendicular to AB, triangle ACQ and triangle ACP are similar triangles by the AA similarity criterion.
Using the similarity, we can set up the proportion:
AC/AP = CQ/CP
Again, since AP : CQ = 3 : 4, we can substitute the value of AP from the previous proportion:
AC / [(3/4) * CQ] = CQ/CP
Simplifying the equation, we get:
AC/[(3/4) * CQ] = 1/CP
Now, we want to find the value of sin A and sin C.
In triangle ABC, sin A = BC/AB
Substituting the value of BC from the proportion AB/AP = BC/BP, we get:
sin A = (AB * BP) / (AB * AP)
Canceling out the common factor AB, we get:
sin A = BP/AP
In triangle ACQ, sin C = CQ/AQ
Substituting the value of AQ from the proportion AQ/CP = AC/[(3/4) * CQ], we get:
sin C = CQ / (CP * [(3/4) * CQ])
Canceling out the common factor CQ, we get:
sin C = 1 / (CP * (3/4))
Now, we have the values of sin A and sin C.
The numerical value of sin A/sin C is:
(sin A) / (sin C) = (BP/AP) / [1 / (CP * (3/4))]
Since BP/AP = 4/3, we can simplify the expression:
(sin A) / (sin C) = (4/3) / [1 / (CP * (3/4))]
To further simplify, we can multiply the numerator and denominator by CP * (3/4):
(sin A) / (sin C) = (4/3) / [ (1 * CP * (3/4)) / (CP * (3/4))]
Canceling out the common factors:
(sin A) / (sin C) = 4/3
Therefore, the numerical value of sin A/sin C is 4/3.