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4:54 PM Thu Jan 16 cdn.assess.prod.mheducation.com Question 3 The angle bisectors of APQR are PZ, QZ, and RZ. They meet at a single point Z. (In other words, Z is the incenter of APQR.) Suppose YZ = 22, QZ = 23, mz WPY 38°, and mzXQZ = 54°. Find the following measures. Note that the figure is not drawn to scale. P W Z X R Y mzXQW WZ = = 0 mz XRZ = 0°

Ja дх dx dx Q3: Define the linear functional J: H()-R by تاریخ (v) = ½a(v, v) - (v) == Let u be the unique weak solution to a(u,v) = L(v) in H₁(2) and suppose that a(...) is a symmetric bilinear form on H() prove that a Buy v) = 1- u is minimizer. 2- u is unique. 3- The minimizer J(u,) can be rewritten under J(u)=u' Au-ub, algebraic form Where A, b are repictively the stiffence matrix and the load vector Q4: A) Answer only 1-show that thelation to -Auf in N, u = 0 on a satisfies the stability Vulf and show that V(u-u,)||² = ||vu||2 - ||vu||2 lu-ulls Chu||2 2- Prove that Where =1 ||ul|= a(u, u) = Vu. Vu dx + fu. uds B) Consider the bilinear form a(u, v) = (Au, Av) + (Vu, Vv) + (Vu, v) + (u, v) Show that a(u, v) continues and V- elliptic on H(2) (3) (0.0), (3.0)

A convex polygon is said to be regular if all of its sides have the same length and all angles between sides are the same. Let Pr denote the regular convex n-sided polygon. Thus, P3 is the equilateral triangle, P₁ is the square, P is the pentagon etc. Compute a formula for the size of any internal angle of Pn.

+ Recall that a map, f: R2 R², is an isometry if |P-Q| = |ƒ(P) — ƒ (Q) for all pairs of points P and Q in R². Thus, f is a distance preserving map. Show that an isometry, f: R² → R² also preserves angles. In other words if two line segments meeting at a point determine an angle a, their image line segments meeting at the image of that point also determine the angle a.

Using Harriott's formula for the area of a triangle on the unit sphere, S2, compute the Euler characteristic X(S2). Make sure to explain what the Euler characteristic is. (The point of the exercise is to show that this notion is well-defined.)