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In triangle ABC, AB has a length of 10 and D is the midpoint of AB. What is the length of line segment DC? (1) Angle C= 90(2) Angle B= 45

Answer: A

From statement (1): it is given that angle C = 90 degrees ...this implies that ABC is a right angle triangle with AB as the hypotenuse and DC as the median. We know that --- In all **right triangles**, the **median** on the **hypotenuse** is the half of the **hypotenuse. Hence DC=5**
In the figure shown, what is the value of x?

(1) The length of line segment of QR is equal to the length of line segment RS (2) The length of line segment of ST is equal to the length of line segment TU

Answer: C

From statement (1): Length of line segment of QR is equal to the length of line segment RS ..this implies angle RQS = angle RSQ = p(say)From statement (2): Length of line segment of ST is equal to the length of line segment TU .. this implies angle TUS = angle TSU = q(say) Hence p+p+angle QRS = 180 --- eq(1) and q+q+angle UTS = 180 --- eq(2)Thus, p+q+x = 180 Now because angle RPT = 90, QRS+UTS= 90 adding eq(1) and eq(2) we get:2p+2q+QRS+UTS = 360 2p+2q+90=360 p+q = 270/2 = 135 Now x = 180-p-q..hence the answer C

x = 180 - (p+q) = 180 - 135 = 45