《2025年暑假训练营学年总复习八年级数学北师大版希望出版社》

1. (宿迁中考)若实数m,n满足等式$|m-2|+\sqrt {n-4}= 0$,且m,n恰好是等腰$\triangle ABC$的两条边的边长,则$\triangle ABC$的周长是 (B)
A.12
B.10
C.8
D.6

答案： B

2. (成都中考)等腰三角形的一个底角为$50^{\circ }$,则它的顶角的度数为$80^{\circ}$.

答案： $80^{\circ}$

3. (邵阳中考)如图,在等腰$\triangle ABC$中,$AB= AC,$$∠A= 36^{\circ }$.将$\triangle ABC$中的$∠A$沿DE向下翻折,使点A落在点C处.若$AE= \sqrt {3}$,则BC的长是__$\sqrt{3}$__.

答案： $\sqrt{3}$

4. (桂林中考)如图,在$\triangle ABC$中,$∠A= 36^{\circ },AB= $$AC,BD平分∠ABC$,则图中等腰三角形的个数是__3__.

答案： 3

5. (北京中考)如图,在$\triangle ABC$中,$AB= AC,∠A= $$36^{\circ },BD平分∠ABC$交AC于点D.
求证:$AD= BC.$
证明：$\because AB = AC$，$\angle A = 36^{\circ}$，$\therefore \angle ABC = \angle C = 72^{\circ}$。$\because BD$平分$\angle ABC$交$AC$于点$D$，$\therefore \angle ABD = \angle DBC = 36^{\circ}$，$\therefore \angle A = \angle ABD$，$\therefore AD = BD$。$\because \angle C = 72^{\circ}$，$\therefore \angle BDC = 72^{\circ}$，$\therefore \angle C = \angle BDC$，$\therefore BC = BD$，$\therefore AD = BC$。

答案： 证明：$\because AB = AC$，$\angle A = 36^{\circ}$，$\therefore \angle ABC = \angle C = 72^{\circ}$。$\because BD$平分$\angle ABC$交$AC$于点$D$，$\therefore \angle ABD = \angle DBC = 36^{\circ}$，$\therefore \angle A = \angle ABD$，$\therefore AD = BD$。$\because \angle C = 72^{\circ}$，$\therefore \angle BDC = 72^{\circ}$，$\therefore \angle C = \angle BDC$，$\therefore BC = BD$，$\therefore AD = BC$。

6. (内江中考)如图,$AD平分∠BAC,AD⊥BD$,垂足为点D,$DE// AC.$
求证:$\triangle BDE$是等腰三角形.

答案：
证明：如图。$\because DE// AC$，$\therefore \angle 1 = \angle 3$。$\because AD$平分$\angle BAC$，$\therefore \angle 1 = \angle 2$，$\therefore \angle 2 = \angle 3$。$\because AD\perp BD$，$\therefore \angle 2 + \angle B = 90^{\circ}$，$\angle 3 + \angle BDE = 90^{\circ}$，$\therefore \angle B = \angle BDE$，$\therefore BE = DE$，$\therefore \triangle BDE$是等腰三角形。

7. (陕西中考)如图,将两个大小、形状完全相同的$\triangle ABC和\triangle A'B'C'$拼在一起,其中点$A'$与点A重合,点$C'$落在边AB上,连接$B'C$.若$∠ACB= $$∠AC'B'= 90^{\circ },AC= BC= 3$,则$B'C$的长为 (A)


A.$3\sqrt {3}$
B.6
C.$3\sqrt {2}$
D.$\sqrt {21}$

答案： A

8. (襄阳中考)已知CD是$\triangle ABC$的边AB上的高,若$CD= \sqrt {3},AD= 1,AB= 2AC$,则BC的长为____$2\sqrt{3}$或$2\sqrt{7}$____.

答案： $2\sqrt{3}$或$2\sqrt{7}$

9. (东营中考)我国古代有这样一道数学问题:“枯木一根直立地上,高二丈,周三尺,有葛藤自根缠绕而上,五周而达其顶,问葛藤之长几何?”题意是:如图,把枯木看作一个圆柱体,因一丈是10尺,则该圆柱的高为20尺,底面周长为3尺,有葛藤自点A处缠绕而上,绕5周后其末端恰好到达点B处.则问题中葛藤的最短长度是__25尺__.

答案： 25尺