Determine if the point (4, -3) is a solution
1 answer:
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Answer:
Given: Two Isosceles triangle ABC and Δ PBC having same base BC. AD is the median of Δ ABC and PD is the median of Δ PBC.
To prove: Point A,D,P are collinear.
Proof:
→Case 1. When vertices A and P are opposite side of Base BC.
In Δ ABD and Δ ACD
AB= AC [Given]
AD is common.
BD=DC [median of a triangle divides the side in two equal parts]
Δ ABD ≅Δ ACD [SSS]
∠1=∠2 [CPCT].........................(1)
Similarly, Δ PBD ≅ Δ PCD [By SSS]
∠ 3 = ∠4 [CPCT].................(2)
But, ∠1+∠2+ ∠ 3 + ∠4 =360° [At a point angle formed is 360°]
2 ∠2 + 2∠ 4=360° [using (1) and (2)]
∠2 + ∠ 4=180°
But ,∠2 and ∠ 4 forms a linear pair i.e Point D is common point of intersection of median AD and PD of ΔABC and ΔPBC respectively.
So, point A, D, P lies on a line.
CASE 2.
When ΔABC and ΔPBC lie on same side of Base BC.
In ΔPBD and ΔPCD
PB=PC[given]
PD is common.
BD =DC [Median of a triangle divides the side in two equal parts]
ΔPBD ≅ ΔPCD [SSS]
∠PDB=∠PDC [CPCT]
Similarly, By proving ΔADB≅ΔADC we will get, ∠ADB=∠ADC[CPCT]
As PD and AD are medians to same base BC of ΔPBC and ΔABC.
∴ P,A,D lie on a line i.e they are collinear.
Answer:
a) See the proof below.
b)
Step-by-step explanation:
Part a
For this case we assume that we have the following series and this series has a finite radius of convergence
and we assume that
for all n, this information is given by the problem.
We assume that the series converges at the point since w eknwo that converges, and since converges we can conclude that:
For this case we need to show that converges also for
So we need to proof that
We can do some algebra and we can rewrite the following expression like this:
and we see that the last series is alternating.
Since we know that converges then the sequence {
} must be positive and we need to have
And then by the alternating series test we can conclude that also converges. And then we conclude that the power series
converges for
,and that complete the proof.
Part b
For this case we need to provide a series whose interval of convergence is exactly (-1,1]
And the best function for this
Because the series converges to
when
using the root test.
But by the properties of the natural log the series diverges at because
and for
we know that converges since
is an alternating series that converges because the expression tends to 0.